# Ultraelliptic Integrals and Two-Dimensional Sigma Functions

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## Abstract

This paper is devoted to the classical problem of the inversion of ultraelliptic integrals given by basic holomorphic differentials on a curve of genus 2. Basic solutions *F* and *G* of this problem are obtained from a single-valued 4-periodic meromorphic function on the Abelian covering *W* of the universal hyperelliptic curve of genus 2. Here *W* is the nonsingular analytic curve *W* = {**u** =(*u*_{1}, *u*_{3}) ∈ ℂ^{2}: *σ*(**u**) = 0}, where *σ*(**u**) is the two-dimensional sigma function. We show that *G*(*z*) = *F*(*ξ*(*z*)), where *z* is a local coordinate in a neighborhood of a point of the smooth curve *W* and *ξ*(*z*) is the smooth function in this neighborhood given by the equation *σ*(*u*_{1}, *ξ*(*u*_{1})) = 0. We obtain differential equations for the functions *F*(*z*), *G*(*z*), and *ξ*(*z*), recurrent formulas for the coefficients of the series expansions of these functions, and a transformation of the function *G*(*z*) into the Weierstrass elliptic function ℘ under a deformation of a curve of genus 2 into an elliptic curve.

## Key words

functions on sigma divisor Abel-Jacobi inversion problem 4-periodic meromorphic functions## Preview

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## Notes

### Acknowledgments

The authors are grateful to A. B. Bogatyrev and V. Z. Enolski^{†} for stimulating discussions and literature references.

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